COMPARISON-THEOREMS FOR THE MATRIX RICCATI EQUATION

Comparison Theorems for the Matrix Riccati Equation

Jonq Juang* and Mei Tzu Lee Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan, R. 0. C.

Submitted by Richard A. Brualdi

ABSTRACT

We derive comparison theorems for the matrix-valued Riccati equations of the form R:(z) = BJz) + A,(.z)R,(.z) + R,(z)D,(z) + R,(z)Ci(z)Ri(z), R&O) = R”,,, i = 1,2. Such equations arise in transport problems and other applications. Sufficient conditions under which R,(z) - R,(z) >, 0 and RI(z) - R,(z) > 0 (in the com- ponentwise sense) for all z are given, respectively. The comparison theorems in which the positivity is defined in terms of positive semidefiniteness were obtained by Royden. Because of the intrinsic disparity in the nature of posititity structure, the techniques developed there cannot be applied to our problem.

1. INTRODUCTION

Consider the matrix-valued Riccati equations of the form

R’(z) = B(z) +

A(z)R(z)

+ R(z)D(z)

+

R(z)C(z)R(z),

(1)

R(0)

=

R,.

We assume that coefficient matrices A, B, C, and D of appropriate dimensions are piecewise continuous on LO, a>.

*The work was partially supported by the National Science Council of the Republic of China.

LINEAR ALGEBRA AND ITS APPLICATIONS 196:183-191(1994) 0 Elsevier Science Inc., 1994

183

Such equations arise in transport theory (see e.g., [l-4,6]) and other applications. We refer to [l], [2], and [6] for further physical background regarding this problem.

Let the matrix E(z) be defined in block form by

E(z)

=

_i

C(z)

A(z) B(z)

D(z)

_i

;

(2)

it is called the matrix of the Riccati equation. If G and H are two real matrices, we write G > H to denote that Gij - H, > 0, for all i,j, i.e., the element in the ith row and jth column of G is no less than that of H. If Gij - H,, > 0 for all i, j, then we write G > H.

In Section 2 we introduce notation and underlying assumptions, and record some well-known results for completeness and ease of reference. Section 4 is devoted to the main results. Specifically, if R, satisfies a Riccati equation of the form (1) with matrix E,(z) and if R, satisfies an equation of the same form with matrix E,(z), then we establish conditions under which R,(z) > R,(z) for all z and R,(z) > R,(z) for all z, respectively. The remainder of this introductory section consists of an overview of previous work related to the present work.

The earlier study most closely related to ours are that of Royden [S]. In [S], the positivity structure is defined in the sense of positive semidefiniteness. Other differences are that in [S] the coefficient matrices are independent of z, and that the matrices B and C are required to be symmetric and A = DT. Perhaps most importantly, because the difference in the posi- tivity structure, the techniques developed in [5] in obtaining comparison theorems cannot be applied to our problem. A different approach needs to be developed.

2. PRELIMINARIES

In the following discussion the matrix functions A, B, C, and D will be supposed to be defined on a given interval [O, m) on the real line, and satisfy the following hypothesis:

A(z), B(z), C(z), and D(z) are matrices of appropriate dimen- sions, which on arbitrary compact subintervals [a, b] of [O, m> are, (3) respectively, piecewise continuous.

MATRIX RICCATI EQUATION 185

NOTATION. Let xE be such that [O, x,> is the maximal interval of

existence of a solution.

It is clear that under this hypothesis the equation (1) with the matrix E(z) has a unique solution on the maximal interval [O, xE) of existence. We furthermore assume that the coefficient matrices satisfy the additional conditions

B(z)

23

0,

C(z)

2 0 for 2 a.e. on [0, xE), _Pa)

A(z)*> *O, D(z) * > *o for z a.e. on [0, xE). _(5b) Here (5b) denotes that A( z> and D(z) h ave nonnegative entries except possibly along their diagonals. Except for a change of notation the following result is essentially Theorem 9.2 of Chapter 2 of Reid [4], and we refer to this source for a proof.

THEOREM 1. Suppose the matrix E(z) of (2) satisfies (3) and (5) and the initial value R, > 0. Then the solution R(z) is unique and has nonnegative entries for z E [O, xE).

In what follows, we shall introduce the concept of irreducibility (see e.g. C71).

DEFINITION 1. A matrix A = (cz,~) of order N is irreducible if N = 1 or

if N > 1 and given any two nonempty disjoint subsets S and T of W := 1L.2,. . . , N, and whose union is W, there exist i E S and j E T such that aij # 0.

An equivalent statement of irreducibility is given in the following:

THEOREM 2 (See e.g. Theorem 2.5.2 of [7]). A matrix A of order N > 1

is irreducible if and only if given any two distinct integers i andj, 1 Q i, j < N,

either aij f 0 or there exist i,, i,, . . . , i, such that aii,ui,i, -.* aij # 0.

3. A COMPARISON THEOREM FOR MATRIX RICCATI

EQUATIONS

Our object in this section is to derive some comparison theorems for matrix Riccati equations of the form (1). We shall assume throughout that

the matrices Ei(z), i = 1,2, of Equation (1) satisfy (3) and (9, and that the initial values R,,, R,, > 0.

THEOREM 3. Let R, and R, be, respectively, solutions of Equation (1) with matrices E,(z) and E2( z), and with initial values R,, and Roz. Suppose that

E,(

z>

G

Ed

z>

(6)

and that

Then R,(z) < R,(z) on [O, b), where b = minix,,, xE,). If, in addition, R,, < R,,, then R,(z) < R,(z).

Proof. Define Ai = A,(z) - diag Ai( fii(z) = Di(z) - diag Di(z), i = 1,2. We h ave immediately, for i = 1,2, that

R:(s) - [diag Ai( s)) Ri( s) - R,(s)[diag oi( S)]

= q(s) + &(s)Ri(s) + Ri(s)fi,i(s) + Ri(s)G(s)Ri(s). (8)

Premultiplying and postmultiplying the equation (8) by the integration factors expj’,” diag Ai dt and explS” diag: Di(t> dt, respectively, and integrating the resulting equation with respect to s from 0 to z, we obtain, for i = 1,2, that

‘diag Ai dt

X exp l’diag Di(t) dt ds.

i s i

MATRIX RICCATI EQUATION 187

Let us define the standard Picard iteration {R(“)(z)} by

(10)

where .Y~ is defined in (9). It is clear, via the assumptions (3) and C5), that gi is a monotone operator. An easy induction gives that

0 <

Ryy

2) <

zp+q

z)

Q Ri(

2)

₍₁₁₎

for every z E [O,b) andeach m = O,l,... . Moreover, it follows readily from the assumptions (6) and (7) that

0 Q

Ryy

2) <

R’,“‘(

2)

(12)

for every .z E [0, b) and each m = 0, 1,. . . . From (11) and (121, we have that R,(z) < R,(z) Vz E [O, b).

The last assertion of the theorem follows from (3), (5), (6), (7), and (9).

This completes the proof of Theorem 3. n

The remainder of this section is devoted to the problem of finding nontrivial sufficient conditions on &(.z) so that one solution is strictly greater than the other.

Let the matrix M(z) be defined by

M(z) = A&) - 4(4 B,(z) - B,(z)

C,(z) - C,(z) a4 - W) = (“ij(4). (13)

For simplicity, we will assume here that the matrices Ai( Bj(.z), C,(z), and D,(z), i = 1,2 are n X n. For each .z > 0, let G(z) be the directed graph on the vertices 1,2,. . . ,2n such that the edge (i,j) f G(z) if and only if mij > 0.

THEOREM 4. Suppose the hypothesis of Theorem 3 holds. Suppose, furthermore, there exists an E > 0 such that for every pair of indices i, n + j,

1 < i, j < n, there is a directed path from i to n + j in G(z) for all z in (0, E). Then R,(z) > R,(z) for z E (0, b).

Proof. Using (9), we obtain that, for z E (0, b),

R2( z) - R,(z) = [

R’,O’( z) - R$“( z)] + /:exp[/i&ig A,(t) dt) s

x[ US) + &(s)R,(s) + R&)%(s) +R,W,(s)R,W]

X exp

‘diag A,(t) dt

By the facts that diag A,(z) < diag A,(z), diag D,(z) < diag D,(z), and R,,(z) < R,,(z) for x E [0, b), we have that

MATRIX RICCATI EQUATION 189

for x E [O, b),

&Lz)C,(z)&?(~) - &(4C,(4%4

a [R,(z) - 4(4lP,(4&(4 - C,(44(41

a I%(4 -

M41 tc,w - Cd41 t %(z) - W)l*

The first inequality above is justified by Theorem 3, the assumption, and the fact that

R,(z)%(z)&(z) + &(z)C,(z)R,(z) 2 2%(z)C,(z)R,(z)* Similarly, we obtain

and

Using the above estimates, we have that

X exp /Zdiag Dl( t) dt

i 0 1

dt .

₍₁₄₎

To complete the proof, it suffices to show that the right-hand side of (13) is greater than 0. To this end, let P = i,, i,, . . . , i, be a sequence of m + 1 distinct vertices between 1 and 2n such that i, = i E U, i, = n + j E T, where U = {1,2,..., n), T = {n + 1, n + 2, . . . ,2n}. We now proceed to

show that (R,(z) - R,(z))~~ > 0 for 1 < i, j < 12, by induction on m, the length of the path P. For each i, j, if m = 1, then, via (14),

CR2tz)

-

RI(Z))ij

a j

LO, zln7(0, E)

exp ‘(diag A,(t))ii dt

x(B~(s)) -

B,(s))ijeq

/‘(diag

ol(t))jjdt

s

> 0.

Suppose the desired result is true for paths of length less than or equal to k, and let P = i,, i,, . . . , ik+l, be a fixed path of length k + 1 with i, = i E u, ik+l =n+jET. If i,= n + I E T, then (R,(z) - R,(z))~, > 0 and

(6,(z) - G,(z)), > 0 for z E [O, b). Thus,

/‘(dial: Di(t))jj dt s

If i, E ,V, the same conclusion follows from a similar estimate applied to the term [A,(s) - A,(s)l[R,(s) -

R,(s)1

occurring on the right side of (13). Finally, if neither i, E T nor i, E V, then necessarily there exists p, 1 < p < k - 1, such that i, = 4 + n E T and i,, 1 = r E U. It follows from the inductive assumption that both (R,(z) - R,(z))~, and (R,(z) - R,(z)),~ are positive on [O, b). Th e d esired conclusion then comes from the estimate

X(%(S) - w-))i&W

-

cmqr

x(R2(s) -

Rl(s))rj Xexp i/ ‘(diag D(t))jj dt s

MATRIX RICCATI EQUATION 191

The following corollary is a direct consequence of Theorems 2 and 4. COROLLARY 1. Suppose M(z), as defined in (131, is irreducible for all z in (0, F), for some E > 0. Then the assertion of Theorem 4 holds.

REFERENCES

R. Bellman, Introduction to Matrix Analysis, 2nd ed., McGraw-Hill, New York, 1970.

R. Bellman and G. M. Wing, An Introduction to Inoariant lmbedding, Wiley, New York, 1975.

G. H. Meyer, Initial Value Methods for Boundary Value Problems, Academic, New York, 1973.

W. T. Reid, Riccati Differential Equations, Academic, New York, 1972.

H. L. Royden, Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math. XLI:739-746 (1988).

G. M. Wing, An Introduction to Transport Theory, Wiley, New York, 1962. D. M. Young, Iterative Solution of Large Linear Systems, Academic, New York, 1971.